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Inequalities in Triangles

  • Lesson
9-12
1
Geometry
Heather Lynn Johnson
Denver, CO

Students will use pasta to create models of triangles and non-triangles and
investigate the relationship between the longest side of the triangle and the sum of the
other two sides of the triangle. In addition, students will measure the sides and angles of a
scalene triangle and investigate the relationship between the location of the largest angle
and largest side in a triangle.

During this lesson, students will engage in activity where they will investigate relationships of sides and angles in a triangle. While students are engaging in the activity, the teacher's role will be that of facilitator. When students complete the activity, the teacher will use the questions on the activity sheet to lead discussion about the important concepts. The lesson flows well when students begin with the activity for the triangle inequality and move to the activity for the inequalities for sides and angles of a triangle.

Provide students with the The Triangle Inequality activity sheet, a ruler, and 8‑10 pieces of thin pasta. Students should work individually on the first five questions on the activity sheet, and they should work with a partner on the last three questions.

pdficon  The Triangle Inequality Activity Sheet 

Some students may not understand what it means to have three sides that cannot make a triangle. In an attempt to create a non‑triangle, students may arrange three sides that form a triangle in such a way that they appear to form a non-triangle. It is important to stress that it must be impossible to form a triangle regardless of how the three pieces are arranged.

When students describe the relationship between the sum of the measures of the small and medium sides and the measure of the large side of the triangles and the non-triangles, some students will describe the sum of the measures of the small and medium sides as "far away from", "close to", or "different from" the measure of the large side. As students begin to make their conjectures in Question 5, encourage them to refine their ideas.

After students have had time to work on Questions 6‑8 with their partner, use these questions to focus class discussion about the triangle inequality. Ask students to share their conjectures, and most importantly why they think their conjecture would work for any triangle. Make certain that students realize that testing a few triangles is not sufficient evidence that a conjecture will work for every triangle. Encourage students to reason about the relationships between the sum of measures of the small and medium sides and the measure of the large side of the triangle. For example, ask students why it is impossible to have a triangle where the sum of the measures of the small and medium sides is less than the sum of the measure of the large side.

When discussing Question 7 on the activity sheet, use pasta to demonstrate why it is impossible for the sum of measures of the small and medium sides to be equal to the measure of the large side of a triangle. Break pasta into two equal parts. Let one part be the large side of the triangle. Break the other part into two pieces (not necessarily equal), and let those be the small and medium sides of the triangle. The small and medium sides collapse onto the large side, creating a line segment, not a triangle.

2339 nontriangle

When discussing Question 8 on the activity sheet, describe how the sum of the measures of any other pair of sides would have to be greater than the measure of the remaining side, because the measure of at least one of the sides in the pair would already be greater than the measure of the remaining side before finding the sum. For example, the sum of the measures of the large and small sides would have to be greater than the measure of the medium side, because the large side is greater than the medium side.

Conclude the triangle inequality part of the lesson by discussing the significance of finding the sum of the measures of the small and medium sides of the triangle. Since the triangle inequality states that the sum of the measures of any two sides of a triangle will always be greater than the measure of the remaining side, by determining that the sum of the measures of the small and medium sides is greater than the measure of the large side, one can be certain that the other inequalities will hold.

To begin the inequalities for sides and angles of a triangle part of the lesson, provide students with the Inequalities for Sides and Angles of a Triangle activity sheet and a protractor. Students should begin by working individually on the Activity Sheet.

pdficon  Inequalities for Sides and Angles Activity Sheet 

While students are working on the activity sheet, circulate around the room to monitor student progress. Students who have not used a protractor on a regular basis can have difficulty measuring angles. Remind students to consider whether or not their three angle measurements are reasonable based on the sum of angles in a triangle. Students may have difficulty understanding what it means for a side to be opposite an angle in a triangle. It may be helpful to draw a diagram, such as the one below.

2339 img1

Moreover, it is important for students to realize that although the longest side of the triangle is opposite the greatest angle, a side opposite an 82°‑angle in one triangle can be greater than the side opposite a 115°‑angle in another triangle. For example, provide students with a diagram such as the one below, and ask students if it is possible for AC > DF, if m∠B = 82°, and m∠E = 115°.

2339 image 2
[Yes, because all that is known for certain is that AC and DF are greater than the measures of any other sides in their respective triangles. There is no way to know how their lengths compare to the side lengths of other triangles.]

In reasoning about Question 7 on the activity sheet, an acceptable argument would be that if a scalene triangle had two congruent angles, then those angles would have to be opposite congruent sides, but since the scalene triangle has no congruent sides this is impossible.

Conclude the lesson by asking students to describe both the triangle inequality and the inequalities for sides and angles of a triangle using words and symbols.

For example:

  • The sum of the measures of any pair of sides in a triangle will always be greater than the measure of the remaining side.
    [s + ml]
  • The longest side of a triangle will always be opposite the greatest angle of the triangle, and the shortest side will always be opposite the smallest angle.
    [Given ΔABC with m∠A > m∠B > m∠C, we know that BC > AC > AB.]

Assessments 

  1. Two sides of a triangle are 6 cm and 10 cm long. Determine a range of possible measures for the third side of the triangle.
    [4 cm < x < 16 cm]

Given that m∠CDB = 65°, m∠CBD = 72°, m∠ADB = 34°, and m∠A = 87°, list the segments in the diagram in order from longest to shortest.

2339 image 3
[CD, CB, BD, AD, AB.]

Extensions 

Discuss the relationship between the triangle inequality and vector addition. Use the following diagram to illustrate:

2339 vectors2

Using the triangle inequality, |x| + |y| ≥ |x+y| .

[Notice that if |x| + |y| = |x+y|, the vectors would be pointing in the same direction and the diagram would not form a triangle.]

Questions for Students 

1. If the sum of the measures of the small and medium sides of the triangle is greater than the measure of the large side of the triangle, why can it be concluded that the sum of the measures of any other pair of sides of the triangle will be greater than the measure of the remaining side?

[The sum of any other pair of sides must include the large side and as such must be larger than either the medium side or the small side by itself.]

2. Is it possible to have a triangle having the sum of the measures of the small and medium sides equal to the measure of the large side?

[If the sum of the small and medium sides is equal to the long side, the triangle will collapse into a single line segment.]

3. The inequality for sides and angles of a triangle states that the longest side of the triangle must always be opposite the greatest angle of the triangle and that the shortest side of the triangle must always be opposite the smallest angle of the triangle. Why would it be impossible to draw a triangle where the longest side of the triangle was not opposite the greatest angle of the triangle?

[Consider a scalene triangle and—without loss of generality—let AB > AC.
Extend side AC and locate point D so that AD = AB.
Triangle ABD is an isosceles triangle, so m∠ABD = m∠ADB. Also, because ∠ACB is an external angle of triangle BCD, m∠ACB = m∠CDB + m∠DBC. Consequently, m∠ACB > m∠CDB = m∠DBA > m∠CBA. Therefore, by the transitive property, m∠ACB > m∠CBA.]

Teacher Reflection 

  • How did the students demonstrate understanding of the materials presented?
  • What were some of the ways that the students illustrated that they were actively engaged in the learning process?
  • Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were these adjustments effective?

Learning Objectives

Students will:

  • Investigate the relationship between the largest side and the sum of the remaining sides in a triangle.
  • Investigate the relationship between the largest side and the largest angle in the triangle.
  • Use the triangle inequality to solve problems involving triangles.
  • Use the inequality for sides and angles in a triangle to solve problems involving triangles.